You know that feeling when you see a word and you know it's a word but it just seems so... odd. I know I've heard both sung and sang before but I can't remember if either one is wrong or if one is used in different times. Sung sang sung sang, they don't seem English to me anymore

Grammatically, an object in active voice becomes the subject when transformed into passive voice.

In "I sang a song," 'a song' is the direct object and the theme of the sentence.

In "A song was sung by me," 'a song' is still the theme of the sentence, but it has become the grammatical subject.

The agent of both sentences is the same (I/me), but in the active, it is the subject and in the passive, it is the object of a preposition.

"It is a song (that is/was) sung by me." The subject is 'it,' 'a song' is the subject-complement, and '(that is/was) sung by me' is a relative clause. The relative pronoun here, 'that', is the subject of the relative clause

"It is a song (that) I sing. Same as the previous sentence, except for 'that,' which is an object in this sentence.

When ～たい is added to the renyoukei (連用形, conjugative form) of a verb, it means "want to ~", but in this case 歌い is the renyoukei. So no, that doesn't apply here.

I don't mean to go too deep into grammar (I recommend getting a proper book for that), but the renyoukei is different depending on the type of verb. With most -let's say "regular"- verbs you can tell by looking at the rentaikei (連体形, so-called 'dictionary form'). If the verb ends in -ku, then the renyoukei is -ki, like 聞く → 聞き. This is where you can add ~ます, but also ~たい.

Yes it would be; "utaimashita" is "I sang" (from the past tense of utaimasu; the polite form of utau).

"utaishimashita" on the other hand is not a verb. It looks like "utai" + "shimashita" (the polite past tense of "to do"). The only noun like that's pronounced "utai" would be 謡, which is the chanting of a Noh play. Granted, that's not too far away from singing (related etymology even), but nevertheless distinctly different.

I'm no expert, but my understanding of it as I've learned it in High School and here on Duolingo is that " takusann " ( たくさん ) basically means " a lot " or " many. "
For example:
" 本をたくさん読みます。" ( Honn wo takusann yomimasu . ) , which means " I read a lot of books. " Takusann ( a lot ) is placed after the direct object, Honn ( book ), has been identified and right before it describes the action, Yomimasu ( to read ).
Hope this helps! :-)

It shouldn't. The "many songs" part, which refers to a noun and a quantity thereof, isn't in the Japanese sentence. I know that you don't really sing other things than songs, so it sounds logical to include it, but in English the sentence works just fine without explicitly mentioning them, so adding it would not be an accurate translation here.